Finite Group Databasebeta

← #26125950

Subgroups — #26125950

Order 12032.

554 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246753C211yesmaximal, characteristic
2#20241729C211yesmaximal, characteristic
3#20241335C214yesmaximal, complemented
4C47×#1459C211yesmaximal, characteristic
5C23×Dic94C411yescharacteristic
6C2×#13094003—21no—
7#15689180—21no—
8#15689043—21no—
9#15689035—21no—
10#15689029—21no—
11#15689003C412yes—
12#15688995C2212yes—
13#15688983C411yescharacteristic
14#15688847—21no—
15#15688847C2212yes—
16C2×C4×D4×C47C2211yescharacteristic
17C2×C22⋊C8×C47—21no—
18C47×#324—21no—
19C47×#323C2212yes—
20C22×Dic94—21no—
21C22×Dic94—21no—
22C22×Dic94—44no—
23C22×Dic94—41no—
24C22×Dic94—41no—
25C23×Dic47C2×C412yes—
26#13094003—44no—
27C2×C4⋊Dic47—21no—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47C2×C412yes—
31C4⋊(C47⋊C8)—41no—
32C22⋊(C47⋊C8)—22no—
33#13093935—41no—
34C47⋊#95—21no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C22×D4×C47C2×C411yescharacteristic
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C22×C8×C47—41no—
41C4×D4×C47—41no—
42C4×D4×C47—41no—
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47C2311yescharacteristic
46C2×C42×C47D411yescharacteristic
47C22⋊C8×C47—22no—
48C22⋊C8×C47—41no—
49C24×C47C22⋊C411yescharacteristic
50C24×C47C22×C411yescharacteristic
51C2×D4×C47—22no—
52C2×D4×C47—22no—
53C2×D4×C47—41no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—42no—
59C22×C4×C47—42no—
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—22no—
66C22×Dic47—22no—
67C22×Dic47—44no—
68C22×Dic47—42no—
69C2×Dic94—24no—
70C2×Dic94—44no—
71C2×Dic94—42no—
72C2×Dic94—42no—
73C2×Dic94—48no—
74C2×Dic94—44no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×Dic94—44no—
78C2×C8×C47—42no—
79C2×C8×C47—82no—
80C4⋊C4×C47—41no—
81C22⋊C4×C47—41no—
82C22⋊C4×C47—41no—
83C42×C47—21no—
84C42×C47—21no—
85C22⋊Dic47—24no—
86C22⋊Dic47—44no—
87C4⋊Dic47—42no—
88Dic47⋊C4—44no—
89C2×C47⋊C8—42no—
90C23×C47—21no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—41no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47#10112yes—
99C23×C47C2×C22⋊C411yescharacteristic
100C23×C47#13112yes—
101D4×C47—44no—
102C2×C4×C47—22no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—81no—
116C2×C4×C47#9911yescharacteristic
117C2×C4×C47#9911yescharacteristic
118C2×Dic47—24no—
119C2×Dic47—42no—
120C2×Dic47—44no—
121C2×Dic47—42no—
122C2×Dic47—44no—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126Dic94—44no—
127Dic94—48no—
128Dic94—44no—
129C8×C47—82no—
130C47⋊C8—82no—
131#11575—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—22no—
135C22×C47—22no—
136C22×C47—42no—
137C22×C47—42no—
138C22×C47—42no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—41no—
142C22×C47—41no—
143C22×C47—81no—
144C22×C47#40811yescharacteristic
145C22×C47C2×#9911yescharacteristic
146C22×C47C2×#10111yescharacteristic
147C4×C47—21no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47AΣL(1,𝔽16)12yes—
153Dic47—44no—
154Dic47—82no—
155#2409—471no—
156#1466—471no—
157#1459D4711yescharacteristic
158#1326—474no—
159C2×C47—21no—
160C2×C47—21no—
161C2×C47—42no—
162C2×C47—42no—
163C2×C47#147211yescharacteristic
164C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
165C2×C47#212311yescharacteristic
166C23×Q8—471no—
167C2×C22⋊Q8—941no—
168C2×C4×D4C2×D4711yescharacteristic
169#432—941no—
170C2×C22⋊C8—21no—
171#395—941no—
172#393—941no—
173#391—941no—
174#386—472no—
175#385—472no—
176#380—471no—
177#324—21no—
178#323—472no—
179#323—941no—
180#323C2×D4712yes—
181C47#1157511yescharacteristic, complemented, Sylow
182C22×Q8—941no—
183C22×Q8—941no—
184C22×Q8—1881no—
185C22×Q8—1881no—
186C22×Q8—1884no—
187C22×D4C4×D4711yescharacteristic
188C23×C4—472no—
189C23×C4C4×D4711yescharacteristic
190C23×C4C22⋊D4711yescharacteristic
191C22×C8—41no—
192C22⋊Q8—1884no—
193C4×D4—41no—
194C4×D4—41no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4—942no—
197C2×C4⋊C4C22⋊D4711yescharacteristic
198C2×C22⋊C4—472no—
199C2×C22⋊C4—942no—
200C2×C22⋊C4D18811yescharacteristic
201C2×C22⋊C4C22×D4711yescharacteristic
202C2×C42D18811yescharacteristic
203C4⋊C8—1881no—
204C22⋊C8—22no—
205C22⋊C8—41no—
206C22⋊C8—942no—
207#97—1881no—
208#95—941no—
209#95—942no—
210#95—942no—
211C24(C2×D47)⋊C411yescharacteristic
212C24C2×C4×D4711yescharacteristic
213C2×Q8—944no—
214C2×Q8—1882no—
215C2×Q8—1882no—
216C2×Q8—1884no—
217C2×Q8—1884no—
218C2×Q8—1888no—
219C2×Q8—1884no—
220C2×Q8—1881no—
221C2×Q8—1881no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—942no—
232C22×C4—942no—
233C22×C4—1884no—
234C22×C4—1882no—
235C22×C4(C2×D47)⋊C411yescharacteristic
236C22×C4(C2×D47)⋊C411yescharacteristic
237C22×C4(C2×D47)⋊C411yescharacteristic
238C22×C4C2×D18811yescharacteristic
239C22×C4C2×C22⋊D4711yescharacteristic
240C2×C8—42no—
241C2×C8—82no—
242C2×C8—1882no—
243C4⋊C4—41no—
244C4⋊C4—1882no—
245C4⋊C4—1884no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—944no—
249C22⋊C4—1884no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23#1309394212yes—
261C23C2×(C2×D47)⋊C411yescharacteristic
262C23#1309403012yes—
263Q8—1884no—
264Q8—1888no—
265Q8—1884no—
266D4—44no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—944no—
282C2×C4—1882no—
283C2×C4—1882no—
284C2×C4—1882no—
285C2×C4—1882no—
286C2×C4—1884no—
287C2×C4—1882no—
288C2×C4—1884no—
289C2×C4#1309393811yescharacteristic
290C2×C4#1309393811yescharacteristic
291C8—82no—
292C8—3762no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22#1568908911yescharacteristic
306C22C2×#1309393811yescharacteristic
307C22C2×#1309394211yescharacteristic
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—3762no—
314C4—1884no—
315C4#1568890912yes—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—42no—
320C2#2024175111yescharacteristic
321C2C2×#1568890911yescharacteristic
322C2#2024504311yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC24×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1459C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC2×C22⋊C4×C47C23
hypercenter#1459D47
hypocenterC47#11575
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20241335C2semidirect
181. C47#11575semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202467532
#202417292
#202413352
C47×#14592
#1157547